The number of directed compact site animals and extrapolation formula of directed percolation probability

We consider a quantity, alpha(n,m) = the number of directed compact site animals with m sites at the height n. We give an explicit expression for alpha(n,m) and calculate the mean width and the variance of width at the height n of directed compact animals exactly. The number of alpha(n,m) is identified with a coefficient appearing in the series expansion for the directed site percolation. An extrapolation formula of directed site percolation probability for the square lattice is obtained using this number. The generating function of alpha(n,m) is calculated and some summation formulae are given.